For any pair of positive integers, $a$ and $b$, you can use Euclid's Algorithm to determine the greatest common divisor, $\gcd{(a, b)}$. The least common multiple is then given by$$ab \over \gcd{(a, b)}$$

With three numbers, $a$, $b$ and $c$ we find the least common multiple $m$ of two of them ($a$ and $b$, say) , and then find the least common multiple of $m$ and $c$.

Euclid's Algorithm is very simple. Example: 208 and 117

$$\DeclareMathOperator{\lcm}{lcm}

\begin{aligned}

208 &= 1 \times 117 + 91 \\

117 &= 1 \times 91 + 26 \\

91 &= 3 \times 26 + 13 \\

26 &= 2 \times 13 \\

\gcd{(117,208)} &= 13 \\

\lcm{(117,208)} &= {117 \times 208 \over \gcd{(117,208)}} = {24336 \over 13} = 1872

\end{aligned}$$